## Description

We compare the estimators introduced throughout Chapters 2-4 using a simulated data set that mimics a single decision point, observational study. The goal of this study is to assess the effectiveness of a fictitious medication developed for the treatment of hypertension (systolic blood pressure, SBP, greater than 140mmHg). The primary outcome of interest is the change in systolic blood pressure after six months of treatment.

The data set comprises one thousand patients ($$n=1000$$) diagnosed as hypertensive. Several baseline patient characteristics are available: SBP0, systolic blood pressure (mmHg); W, weight (kg); K, potassium level (mg/dl); Cr, creatinine level (mg/dl); and Ch, total cholesterol (mg/dl). Each patient in this study either received the new medication ($$A=1$$) or received no treatment ($$A=0$$) based on patient/physician discretion. Six months after the treatment decision, the systolic blood pressure of each patient was remeasured, SBP6. The observed data are independent and identically distributed

$\{\text{SBP0}_i, \text{W}_i, \text{K}_i, \text{Cr}_i, \text{Ch}_i, A_i, \text{SBP6}_i\},~ \text{for}~i = 1, \dots, n.$

The primary outcome of interest is $$Y = \text{SBP}0-\text{SBP}6$$, where larger values are considered better. We assume that all participants adhered to the treatment plan to which they were assigned and that no participants dropped out of the study.

For the simulation scenario used to generate the data, the true treatment effect is $$\delta^{\text{*}} = 17.4$$. This scenario implies the following true outcome regression relationship

$Q_{1}(h_{1},a_{1}) = -15 - 0.2~ \text{Ch} + 12~ \text{K} + a_{1}~(-65 + 0.5~ \text{Ch} -5.5~ \text{K}),$

and propensity

$\pi_{1}(h_{1}) = \frac{\exp(-16.348 + 0.0748~\text{SBP0} + 0.017~\text{Ch})}{1+\exp(-16.348 + 0.0748~\text{SBP0} + 0.017~\text{Ch})}.$

The data generating models and the R implementation are provided under the Simulation Scenario heading in the sidebar.

R Environment

Once downloaded, the data set can be loaded into R using

dataSBP <- utils::read.csv(file = 'path_to_file/hyper.txt', header = TRUE)

where ‘path_to_file’ is the full path to the downloaded data set file.

Examining the first few records of the data set

utils::head(x = dataSBP)
  SBP0     W   K  Cr    Ch A SBP6
1  148 101.2 3.7 0.6 167.7 0  154
2  164  50.3 4.4 0.9 174.4 0  161
3  164  70.6 3.7 0.7 202.2 1  160
4  164  65.8 4.1 1.0 297.2 0  191
5  173  84.4 3.5 0.8 242.6 1  155
6  165  66.5 3.6 0.9 254.7 1  145

shows that our data set contains the expected covariates: SBP0, systolic blood pressure measured at baseline; W, weight; K, potassium level; Cr, creatinine level; Ch, total cholesterol; A the treatment received; and SBP6 systolic blood pressure measured six months post treatment.

We must define the outcome of interest, $$Y = \text{SBP}0-\text{SBP}6$$, in the R environment.

y <- dataSBP$SBP0 - dataSBP$SBP6

Note that larger values of the response are better; we seek to decrease patients’ systolic blood pressure.

Remark: $$y$$ is referred to as ‘the response’ or ‘the outcome of interest’ interchangeably in R documentation and the literature. However, we will always refer to $$y$$ as the outcome or outcome of interest to avoid confusion with the nomenclature adopted in later chapters.

## Simulation Scenario

The data for this simulated trial were generated from the following models:

set.seed(seed = 1234L)
n <- 1000L

Baseline Systolic Blood Pressure

$\text{SBP0} \sim \mathcal{N}(\mu=160, \sigma^2=12^2)$

and must satisfy the following constraint: $$140 \mathrm{mmHg} \lt \text{SBP}_{0} \le 200 \text{mmHg}$$.

  # Probability that N(160,144) is below 140
pSBP_L <- stats::pnorm(q = 140.0, mean = 160.0, sd = 12.0)

# Probability that N(160,144) is below 200
pSBP_U <- stats::pnorm(q = 200.0, mean = 160.0, sd = 12.0)

# Generate uniform random values within these limits
prob <- stats::runif(n = n, min = pSBP_L, max = pSBP_U)

# Invert CDF to obtain SBP0
SBP0 <- stats::qnorm(p = prob, mean = 160.0, sd = 12.0)
SBP0 <- round(x = SBP0, digit = 0L)

Weight

Participants’ weights were obtained from height and body mass index, where height (H)

$\text{H} \sim \mathcal{N}(\mu=1.7, \sigma^2=0.1^2)$

and body mass index (BMI)

$\text{BMI} \sim \mathcal{N}(\mu=25.0, \sigma^2=3.5^2);$

$$\text{W} = \text{BMI}*\text{H}*\text{H}$$.

  ## Generate weight

# Generate n normally distributed heights
height <- stats::rnorm(n = n, mean = 1.7, sd = 0.1)

# Generate n normally distributed BMI values
bmi <- stats::rnorm(n = n, mean = 25.0, sd = 3.5)

# Calculate weight
W <- bmi * height * height
W <- round(x = W, digit = 1L)

Potassium Levels

$\text{K} \sim \mathcal{N}(\mu=4.2, \sigma^2=0.35^2).$

  ## Generate potassium levels

K <- stats::rnorm(n = n, mean = 4.2, sd = 0.35)
K <- round(x = K, digit = 1L)

Creatinine Levels

Creatinine levels where generated in a similar manner to $$\text{SBP0}$$,where $$0.4 \le \text{Cr} \le 1.25$$ and

$\text{Cr} \sim \mathcal{N}(\mu=0.82,\sigma^2=0.1^2).$

  ## Generate Creatinine levels

# Probability that N(0.82,0.01) is below 0.4
pCr_L <- stats::pnorm(q = 0.4, mean = 0.82, sd = 0.1)

# Probability that N(0.82,0.01) is below 1.25
pCr_U <- stats::pnorm(q = 1.25, mean = 0.82, sd = 0.1)

# Generate uniform random values within these limits
prob <- stats::runif(n = n, min = pCr_L, max = pCr_U)

# Invert CDF to obtain Cr
Cr <- stats::qnorm(p = prob, mean = 0.82, sd = 0.1)
Cr <- round(x = Cr, digit = 1L)

Total Cholestorol

$\text{Ch} \sim \mathcal{N}(\mu=211, \sigma^2=45^2)$

  ## Generate total cholosterol

Ch <- stats::rnorm(n = n, mean = 211.0, sd = 45.0)
Ch <- round(x = Ch, digit = 1L)

$A \sim B\{n=1, p= \pi_{1}(x_{1})\}$

$\pi_{1}(x_{1}) = \text{expit}(-16.348 + 0.078~\text{SBP0} + 0.017~\text{Ch})$ and $$\text{expit}(u) = e^{u}/(1+e^{u})$$.

  ## Generate treatment variable

probXB <- -16.348 + 0.078*SBP0 + 0.017*Ch
prob <- exp(x = probXB) / (1.0 + exp(x = probXB))
A <- stats::rbinom(n = n, size = 1L, prob = prob)

Six-month Systolic Blood Pressure

$\text{SBP6} = \text{SBP0} - \mathcal{N}(\mu=\mu_{SBP6},\sigma^2=3^2).$

$\mu_{SBP6} = -15.0 - 0.2~\text{Ch} + 12.0~\text{K} + A (-65.0 + 0.5~\text{Ch} - 5.5~\text{K})$

  ## Generate change in Systolic Blood Pressure

mn <- {-15.0 - 0.2*Ch + 12.0*K} + A*{-65.0 + 0.5*Ch - 5.5*K}
y <- stats::rnorm(n = n, mean = mn, sd = 3.0)
y <- round(x = y, digits = 0L)

## Systolic Blood Pressure at six months
SBP6 <- SBP0 - y

The true treatment effect is $$\delta^{\text{*}} = -65.0 + 0.5*211 - 5.5*4.2 = 17.4$$ mmHg.

## Summary

The outcome of interest, $$Y = \text{SBP0}-\text{SBP6}$$, is plotted below for patients that did not receive treatment (black) and for those that received the new medication (red). Recall that we define the outcome of interest such that larger values are more desirable. The change in systolic blood pressure, $$Y = \text{SBP0} - \text{SBP6}$$. Red indicates that the patient received $$A=1$$; black indicates that the patient received $$A=0$$.

The summary data for each covariate is given below.

summary(object = dataSBP)
      SBP0             W                K              Cr               Ch              A              SBP6
Min.   :140.0   Min.   : 37.50   Min.   :3.10   Min.   :0.5000   Min.   : 83.0   Min.   :0.000   Min.   : 97.0
1st Qu.:153.0   1st Qu.: 63.30   1st Qu.:4.00   1st Qu.:0.8000   1st Qu.:178.8   1st Qu.:0.000   1st Qu.:146.0
Median :161.0   Median : 71.90   Median :4.20   Median :0.8000   Median :208.3   Median :0.000   Median :158.0
Mean   :161.6   Mean   : 72.65   Mean   :4.21   Mean   :0.8191   Mean   :209.2   Mean   :0.454   Mean   :157.4
3rd Qu.:169.0   3rd Qu.: 80.90   3rd Qu.:4.40   3rd Qu.:0.9000   3rd Qu.:240.5   3rd Qu.:1.000   3rd Qu.:168.0
Max.   :197.0   Max.   :135.20   Max.   :5.10   Max.   :1.1000   Max.   :332.4   Max.   :1.000   Max.   :214.0  
In the figure below, we show the outcome of interest, $$Y$$ plotted against each baseline covariate. Outcome of interest plotted against each baseline covariate. Red indicates that the patient received $$A=1$$; black indicates that the patient received $$A=0$$.